Why does the energy of the mechanical wave depend on frequency but the EM wave does not? Are there any implications?
In the simple mechanical wave on a string , the energy transmitted is given by:
As a sinusoidal wave moves down a string, the energy associated with one wavelength on the string is transported down the string at the propagation velocity $v$. From the basic wave relationship, the distance traveled in one period is $vT = λ$, so the energy is transported one wavelength per period of the oscillation.
The power in electromagnetic waves is
and the frequency is averaged out. Only the amplitude is explicit.
The classical electromagnetic wave , though, emerges from a huge number of photons with energy $h\nu$. The amplitude is built up by these photons, so the dependence to the frequency is there but not explicit. For mechanical waves the amplitude is a height in three dimensions whereas for electromagnetic waves it is the fields that vary with a frequency as the wave propagates.
In the mechanical wave it is the potential and kinetic energy that alternate and if one goes to the equations for the vector potential A of the electromagnetic field, the equations for mechanical waves and electromagnetic waves become similar, as is shown here, at page 5.
The intensity of an electromagnetic wave is indeed indirectly dependent on frequency. I say indirectly, because energy from EM waves are explicitly only dependent on amplitude. However, the amplitude itself is dependent on individual photons, whose energy is dependent on frequency. The only reason frequency is not in equations describing the energy from an EM wave is because an enormous amount of photons build up to correspond to an amplitude.